Inverse of a Matrix

Q.

Let $A$ be a $3\times 3$ matrix such that $\text{adj}A=\left[\begin{array}{ccc}2& -1& 1\\ -1& 0& 2\\ 1& -2& -1\end{array}\right]$and $B=\text{adj}(\text{adj}A)$. If $\left|A\right|=\lambda$and $\left|{\left(B^{-1}\right)}^T\right|=\mu$, then the ordered pair, $\left(\right|\lambda |,\mu )$ is equal to:

1. $\left(9,\frac{1}{81}\right)$

2. $\left(9,\frac{1}{9}\right)$

3. $\left(3,\frac{1}{81}\right)$

4. $\left(3,81\right)$

Q. If $B=\left[\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right]$ is the inverse of a $3\times 3$ matrix $A,$ then the sum of all values of $\alpha$ for which $det\left(A\right)+I=0,$ is

1. $2$
2. $1$
3. $0$
4. $-1$

Q. If A and B are square matrices of the same order such that AB = BA, then show that (A + B)² = A² + 2AB + B².

Q. Let $A,B,C,D$ be (not necessarily square) four matrices such that $A^T=BCD;$ $B^T=CDA;$ $C^T=DAB$ and $D^T=ABC.$ If $S=ABCD,$ then

1. $S=S^3$
2. $S=S^4$
3. $S=S^2$
4. $S=S^6$

Q.

Let $z=\frac{(11-3i)}{(1+i)}$ If $a$ is real number such that $z-ia$ is real, then the value of $a$ is:

1. $4$

2. $-4$

3. $7$

4. $-7$

Q. Let $A=\left[a_{ij}\right]$ be a real matrix of order $3\times 3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3.$ Then, the sum of all the entries of the matrix $A^3$ is equal to:

1. $1$
2. $3$
3. $2$
4. $9$

Q. If $A$ is an $3\times 3$ non–singular matrix such that $AA\text{'}=A\text{'}A$ and $B=A^{-1}A\text{'}$, then $BB\text{'}$ equals:

1. $I+B$
2. $I$
3. $B^{-1}$
4. $(B^{-1}{)}^{\prime }$

Q.

The coefficient of x⁴⁹ in the product

(x - 1)(x - 3) ... (x - 99) is

(a) - 99² (b) 1 (c) - 2500 (d) - 50

Q.

If $A$ is a skew-symmetric matrix of order $n$ and $C$ is a column matrix of the order of $n\times 1$, Then $C^{T}AC$ is

1. an identity matrix of order $n$

2. an identity matrix of order $1$

3. a zero matrix of order $1$

4. None of these

Q. Let M be a $2\times 2$ symmetric matrix with integer entries. Then, M is invertible if

1. the first column of M is the transpose of the second row of M
2. the second row of M is the transpose of the first column of M
3. M is a diagonal matrix with non-zero entries in the main diagonal
4. the product of entries in the main diagonal of M is not the square of an integer

Q. If $A$ and $B$ are square matrices of the same order and $A$ is non-singular, then for a positive integer $n,(A^{-1}BA{)}^n$ is equal to

1. $A^{-n}{B}^n{A}^n$
2. $A^n{B}^n{A}^{-n}$
3. $A^{-1}{B}^{n}A$
4. $n\left(A^{-1}BA\right)$

Q. Let $P$ be a non-singular matrix such that $I+P+P^2+\cdots +P^n=O$. Then $P^{-1}$ is equal to

1. $P^{n-1}$
2. $P$
3. $-P^n$
4. $P^n$

Q. If $A$ =$\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right],\text{find}{A}^{-1}.$ Hence using $A^{-1}$ solve the system of equations
$2x-3y+5z=11,3x+2y-4z=-5,x+y-2z=-3.$

Q. If $A$ is a square matrix, then $(adjA{)}^{-1}=adj(A^{-1})=$

1. $A^{-1}$
2. $\frac{A}{|A{|}^2}$
3. $\frac{A}{|A|}$
4. $A|A|$

Q. If A is a diagonal matrix of order 3 $\times$ 3 is commutative with every square matrix of order 3 $\times$ 3 under multiplication and trace (A) = 12, then

1. $\left|A\right|=64$
2. $\left|A\right|=16$
3. $\left|A\right|=12$
4. $\left|A\right|=0$

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $A$ is equal to

1. $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
2. $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
3. $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$
4. $\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$

Q. If $A$ is a matrix such that $A^2+A+2I=0,$ then which of the following is/are true?

1. $A$ is non-singular
   
2. $A$ is symmetric
   
3. $A$ cannot be skew-symmetric
   
4. $A^{-1}=-\frac{1}{2}(A+I)$

Q.

$3x=2x+18$, Find the value of $x$

Q. Let $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$ and $10B=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& a\\ 1& -2& 3\end{array}\right].$
If $B$ is the inverse of $A$, then the value $a$ is:

Q. Matrices A and B will be inverse of each other only if A. AB = BA C. AB = 0, BA = I B. AB = BA = 0 D. AB = BA = I

Q. If p + q + r = 0 = a + b + c, then the value of the determinant
$\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ is

1. pq + qb + rc
2. 1
3. none of these
4. 0

Q.

The sum $\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$ for $i\in [0,m]$, where $(p,q)=0,$ if $p>q$ is maximum when $m$ is

1. $5$

2. $10$

3. $15$

4. $20$

Q. If the value of the determinant $\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, then

1. $abc<-2$
2. $abc>0$
3. $abc>-8$
4. $abc=0$

Q. Let $S$ be the set which contains all possible values of $l,m,n,p,q,r$ for which
$A=\left[\begin{array}{ccc}{l}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$ be a non singular idempotent matrix. Then the sum of all the elements of the set $S$ is

Q. Contrapositive of the statement "If a quadrilateral is a square, then it has two pairs of parallel sides".

1. If a quadrilateral has two pairs of parallel sides, then it is a square.
2. If a quadrilateral is not a square, then it does not have two pairs of parallel sides
3. If a quadrilateral does not have two pairs of parallel sides, then it is not a square.
4. If a quadrilateral is a square, then it does not have two pairs of parallel sides

Q.

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}3& -1\\ -4& 2\end{array}\right]$

Q.

Find the value of ${\left(-2\right)}^3\times {\left(-10\right)}^3$

Q. Let $Q=\left(\begin{array}{cc}\cos \frac{\pi }{4}& -\sin \frac{\pi }{4}\\ \sin \frac{\pi }{4}& \cos \frac{\pi }{4}\end{array}\right)$ and
$x=\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$ then$Q^{3}x$ is equal to

1. $\left(\begin{array}{c}0\\ 1\end{array}\right)$
2. $\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$
3. $\left(\begin{array}{c}-1\\ 0\end{array}\right)$
4. $\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right)$

Q. If A and B are any two different square matrices of order n with A – B is non-singular $A^3=B^3$ and A(AB) = B(BA) , then

1. $A^2+B^2=O$
2. $A^2+B^2=I$
3. $A^2+B^3=I$
4. $A^3+B^3=O$

Q.

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$